Reconstructing mechanistic models of alpine basins hydroclimatic behaviour using observed data

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1 Hydrology 6 Reconstructing mechanistic models of alpine basins hydroclimatic behaviour using observed data Paolo Perona 1 and Paolo Burlando Institute of Environmental Engineering, ETH Zurich, Switzerland Abstract. Time series of mean daily values of precipitation p, temperature T and river runoff Q for a glacio-pluvial basin are analyzed with the purpose of obtaining a 2-D differential model describing the hydro-climatic basin behaviour at a seasonal scale. This is done without claiming any classic empirical link between the volume V of water that is stored on the basin and the corresponding river runoff Q. Such a relationship is directly obtained from observed basin data in two steps. We first propose a differential input-output model of the state variables (V, Q) having considered both the physics of the system and the link among the observed quantities (p, T, Q). Second, we extract the numerical value of the unknown model coefficients by analyzing the traectory in the state space (V, Q) using the Traectory Method for the reconstruction of differential equations from time series. The nonlinear model that is obtained mimics well the original data, and seems to catch some essential properties of the underlying system dynamics. Moreover, it appears to be robust enough against forcing, and is thus able to describe the basin dynamics at daily and weekly time scales reasonably well. Results show the benefit of this approach not only to study the linear vs non-linear role played by the different terms of the model, but also to investigate the long-term system behavior under different forcing scenarios. 1. Introduction The hydrological regime of alpine basins is influenced by many climatic (e.g., precipitation, temperature, solar radiation, wind, etc.) and geomorphological (e.g., topography, soil properties, orientation, etc.) factors, whose interrelated actions determine the evolution of the hydrologic variables (snowdepth, river runoff, evaporation, evapotranspiration, etc.). The resulting dynamics are complex: the input of precipitation is transformed into the streamflow output via a mechanism of accumulation, successive ablation, and runoff, which all occur at different spatial and temporal scales. Through this sequence the irregular input of precipitation is eventually transformed into a more regular output on which high frequency oscillations suggest the presence of hourly and daily time scale phenomena (Figure 1). At such short time scales, accounting for the role of all the involved variables may be critical for a correct modeling approach. This requires detailed knowledge about the abovementioned variables, and a simulation approach involving appropriate models such as the Temperature index (Ohmura 1) or the Energy-Mass balance (Corripio and Purves ) ones. Notwithstanding being computationally demanding, these models are particularly useful in predicting the availability of fresh water at hourly and daily time scale. This makes them particularly useful for short-term forecasting. Unfortunately, not that much knowledge is 1 Tel: 41() perona@ifu.baug.ethz.ch Hydrology

2 Perona and Burlando Precipitation [mm] Climate, Hydrology, Topography, Soil properties, etc. Streamflow [m 3 /s] Months Figure 1. Precipitation river runoff transformation for an alpine basin (Mean of 14 years, Runoff measured at Tavagnasco station, Aosta valley, Northern Italy) presently available to model routing phenomena across glaciers, snowpack, and the watershed itself, hence making such models still limited. At the long term, the increased uncertainness in climatic and hydrological variables introduces further noise in the system dynamics. As a result, annual temperature variations become by far the main driving actions influencing the occurrence of storage and the late release of water through the watershed at monthly and seasonal time scales. Figure 1 shows, in this respect a more lumped interpretation of the whole basin dynamics. In turn, this questions the necessity of using detailed models when considering long term evolution. The idea that a fundamental link among only few representative variables is sufficient to capture the essence of the interannual dynamic is therefore appealing. Although not useful at shorter time scales, simplified physical models can have a clear structure that allows enough understanding of the system dynamics in the whole (see, for instance Saltzmann (1983)). Lumped models can in principle perform equally well provided that an intimate link between the main variables of the real system is represented. A similar idea is pursued in this work, and we use the Valle d Aosta region located in Northern Italy as an example of an alpine basin dominated by a glacio-pluvial dynamic. Despite referring to this specific case, our idea is of larger applicability and will be soon extended to other basins. Indeed, thanks to the inductive approach being used we do not need to claim any classic formulation for the precipitation-runoff transformation. This is the interesting and appealing novelty. That is, we accomplish the reconstruction of the seasonal system dynamics by using a recently proposed system identification technique for the reconstruction of differential equations from time series (Eisenhammer et al. 1991, Perona et al. ). This method in principle allows to adapting whatever nonlinear differential models linear in the parameters to the phase space traectory of the measured data. We transfer the potentials of such a technique to reconstruct a semi-empirical differential system driven by two exogenous variables. The reconstructed model can infer important insights on the type and role of the nonlinearities that lump input and outputs. Moreover, it allows us to study the dynamical characteristics of the system and its response to changes in the driving variables (Perona et al. ). 2. Data The Valle d'aosta region is principally a mountainous area with a surface of 3313 Km 2 and a mean elevation of 8 m above sea level. The glacial area

3 Reconstructing mechanistic models of alpine basins behaviour using observed data covers.7% of the entire surface, hence contributing to the glacio-pluvial hydrological regime. The climate of the region and the soil characteristics reduce to a minimum the water loss due to evaporation and infiltration. Hence, solid and liquid precipitations are almost entirely conveyed by the Dora Baltea river, which runoff is measured at the basin closure section, being this latter located in Tavagnasco (263 m above the sea level). Some small reservoirs are present in the valley, but their relevance to the streamflow hydrograph of the Dora Baltea river is practically negligible. The whole dataset accounted for 31 years of continuous spatial measurements of precipitation (mm/day of water equivalent in 22 gauging stations), temperature (Celsius degrees in 8 gauging stations) and streamflow (m 3 /s, in Tavagnasco). This data is available at a daily time scale in the period 1/1/191-12/31/1981. The spatial distribution of the hydrological stations gave a reliable representation of both the daily pluviometric and thermometric regimes of the whole basin. A first data analysis was made for all the variables. Data from the leap year were redistributed in order to reset all time sequences to have 36 data points. Precipitation and temperature were spatially averaged, and a time average of the year-to-year daily data was then made in order to obtain a representative set of the typical interannual basin behavior. Eventually, in order to extract the seasonal component that characterizes precipitation, a moving average with a running window equal to 91 days was applied to the data. This window has been proved to be an optimal compromise to extract the seasonal component without smoothing the series too much (Saltzman 1983). The resulting typical annual behaviour is shown in Figure 2. We call this set of data the training set. Notice the typical periodicity characterizing both temperature and streamflows. The central part of the year shows an evident maximum dictated by the warmer season. In such basins, temperature drives the melting of the snow and is responsible of the earlier maximum of river runoff. Melting of ice, which represents the old water, occurs later and tends to bring the flow in phase with temperature. Also, is worth noting the behavior of the autumnal river runoff when temperatures are still above zero. 2 Q ( 6 ) m 3 /day T ( C) P ( 6 ) m 3 /day 2 Q,T,P Q [ 6 m 3 /day] P [ 6 m 3 /day] 6 - T ( C) Figure 2. Mean annual hydro-climatic behaviour of the Valle d Aosta basin. (Left panel) The training data set of precipitation (p), temperature (T) and streamflow (Q). (Right panel) Corresponding traectory in the state space of three variables p,t,q: arrows indicate evolution. 1

4 Perona and Burlando 3. Reconstructing differential equations from time series We now build up a 2-D nonlinear differential model forced by two exogenous variables (Figure 3). The aim is to obtain for each equation a simple polynomial form involving the hydrological variables up to a certain algebraic degree. Both p(t) and T(t) are considered as being the independent forcing actions. We start by expressing the volume V and the runoff Q in the outlet section as, V = V ( p, Q) Q = Q( T, V ). (1a,b) By differentiating the previous equations and taking advantage of the continuity equation for the basin we have dv = p Q dq Q = T V dt Q V T dv. (2a,b) In Eq. 2a we deliberately neglected contributions by evaporation, evapotranspiration and deep infiltration. An explicit form for the second equation can be obtained by considering the data plots in the planes (Q,T) and (Q,V) and assuming a possible dependence on the two variables V and T. Note that the two equations are coupled by a physical condition that assures the continuity of mass over the domain. While details are reported elsewhere (Perona et al., in preparation), here we show the mathematical form of the resulting differential model dv = p Q dq = c V 2 1 c2vq c3vp c4vt c p c6vt c7q. (3a,b) This is a nonlinear model in the state variables (V, Q), parsimonious in the number of involved coefficients c i, and forced by the presence of the exogenous variables p(t) and T(t). The second equation accounts for nonlinear effects in the product of the two state variables, as well as non-linearities that are im Precipitation input V& = V ( p, Q) Q& = Q( T, V ) Streamflow output Temperature input Figure 3. Conceptual scheme of the input-output model being proposed. Annual precipitation (in water equivalent) and temperature evolutions are the two exogenous variables, which together force the differential model in order to produce the streamflow output. 2

5 Reconstructing mechanistic models of alpine basins behaviour using observed data plicitly contained in the non-homogeneous terms. Such an equation therefore represents a precipitation-storage-runoff transformation in the form of a set of coupled nonlinear ordinary differential equations. Model coefficients have been estimated using the Traectory Method (Eisenhammer et al. 1991), which has the peculiarity of considering at the same time the short and long-term behavior of the system traectory in the corresponding phase space (see also, Perona et al. ). First of all, the phase space traectory of Figure 2 has been interpreted as being periodic and a sequence of years was built. This step, as well as the presence of some irregularity in the data are important for the learning procedure, and make the extracted model more robust against forcing (Perona et al. ). After having chosen several initial conditions x r (V, Q ) on the phase space traectory and fixed a trial set of coefficients c i, the method uses Equations (3) and the values of the exogenous variables (p(t), T(t)) to obtain an estimate of the successive states of the system x m (t t l ) (V(t t l ), Q(t t l )) where t l =h 2(l-1), with l=1,..., l max. In this way a quality function W can be constructed, W = max lmax = 1 l = 1 x ( t t ) x ( t t ), (4) m l r l where is the euclidian norm and x m, x r are the modeled and real states. The parameter t l represents the extent to which the medium and long-term behavior of the system is considered starting from max initial conditions. Thanks to the already know form of the continuity equation (3a), the problem reduces here to the scalar form W = max lmax = 1 l = 1 t t tl Q & m ( τ ) dτ Q m ( t ) Q r ( t t l ). () The optimum values of the coefficients c i are obtained by minimizing the quality function () using the least-square method, i.e. iteratively looking for the condition Wmin = min c W in the linear space of the coefficients. The reader is i referred to Perona et al. () for details. 4. Results and model performances The reconstruction technique was performed using different combinations of the method s parameters (i.e., max, l max, d), and starting with an initial condition for the reconstructed stored volume equal to 7 7 m 3. This volume was roughly estimated based on the current snow and ice reserves of the Valle d Aosta region. Figure 4 shows a compilation of reconstruction sequences performed on the training set. The mechanism with which the model learns about the system dynamics is thus evident. In panel (a) the trial set of coefficients gives an unstable model, regardless of the initial condition from which the model starts on. After a few iterations the current set of coefficients already al- 2 3

6 Perona and Burlando lows for a better approximation of the phase space traectory (panel b). Things improve with successive iterations (panel c); the procedure is then 2 (a) 2 (b) Q [ 6 m 3 /day] Q [ 6 m 3 /day] V [ 7 m 3 ] V [ 7 m 3 ] 2 (c) 2 (d) Q [ 6 m 3 /day] Q [ 6 m 3 /day] V [ 7 m 3 ] V [ 7 m 3 ] Figure 4. Reconstruction of model s coefficients via successive iterations. (a) trial set of coefficients at the first run, where the model s behaviour (broad line) is evidently different from the observed one (thin line). (b) Fifth iteration, (c) tenth and (d) the fiftieth one when the procedure was stopped. stopped as soon as coefficients do not appreciably change with further iterations (panel d).the scenario described by Figure 4 is essentially the same when the procedure parameters are changed. Only minor differences would result, for instance in the extent of the model traectory (i.e, shorter or longer depending on the parameter l max ), the number of initial conditions (i.e., decided upon changing max ), and the frequency with which this latter are chosen (i.e., parameter d that fixes the distance between them). The reconstruction is likely to be successful within a specific range of such parameters (see, Perona et al. () for details). From each reconstruction, seven coefficients were obtained; their average is shown in Table I. The model with the coefficients of Table I reproduces the training set of data well (Figure a,b). While the shape of the mean annual behaviour of streamflow is well reproduced, some of the residual oscillations that still characterize it are instead automatically smoothed. That is, the model catches the main behaviour underlined in the data and, at the same time, seems to endure high frequency oscillations. Unlike some previous efforts (see, Perona et al ), a considerable advantage of inferring the structure of the model since the beginning is the intrinsic robustness that the model develops against dynamical noise. This indicates the possibility of reproducing the basin behavior not only 4

7 Reconstructing mechanistic models of alpine basins behaviour using observed data Table 1. Averaged numerical value of the model coefficients corresponding to the best reconstruction performed with different parameters ( max, l max ). c 1 c 2 c 3 c 4 c c 6 c (a) Q ( 6 ) m 3 /day T ( C) P ( 6 ) m 3 /day (b) V ( 7 ) m 3 Reconstructed 7 Q, T,P V (c) Original ( 6 ) m 3 /day Reconstructed 4 4 (d) Original ( 6 ) m 3 /day Reconstructed 2 3 Q Q Figure. Comparison between the observed data (precipitation smoothed with a 91 day moving average), and the modeled one. (a) Snapshot of the three time series of precipitaton, temperature and streamflow and related comparison for this latter; (b) Reconstructed volumes on the basin. Model behaviour when used to mimic daily streamflow starting from more or less filtered data of precipitation and temperature: (c) 31 days moving average and (d) original daily data. when inputs are not periodic, but also when they contain natural oscillations as those appearing at weekly and daily time scales. This is shown in Figures c,d: three consecutive years of simulation and the comparison with observations are shown in panel c and d. These results come from using either smoothed (31 days m.a.) or original daily data of precipitation and temperatures as new inputs to the model. The comparison is satisfactory, albeit some significant differences are still evident for large events. However, the purpose of such model is not to have a tool for short terms prediction, but to have a determinist tool, which is robust enough against forcing, and is thus able to infer some clues about the system behaviour at a longer term. The model represents a dissipative nonlinear dynamical system as can be seen by rewriting the equation as a canonical non-homogeneous 2 nd order ODE (Perona et al., in preparation). This system has a periodic attractor (i.e., a limit cycle of period one) when forced with the training data set. Viceversa, for hypothetic vanishing precipitation, the system possesses an invariably stable equilibrium point at zero. The high sensitivity of such systems versus precipi-

8 Perona and Burlando tation (Dingman 2), allows for other scenarios where periodic or quasiperiodic attractors can be obtained depending on the forcing characteristics and frequency (Perona et al., in prepar.). The possibility that transition to chaos occurs is also being investigated, as well as the effects of trends in the inputs. Figure 6 shows, for instance, the transitory induced by a weak linear trend in the historical data (here positive for temperature and (a) 2 (b) 7 V 6 Q Figure 6. Effects of linear trends in the inputs in the resulting volume (a) and streamflow (b) evolution. In this case for an increase of 2 degrees over 3 years and a % reduction in precipitation volume, the model foresees a % reduction of the stored volume. negative for precipitation) on the stored volumes and the corresponding river runoffs. Further analyses are being explored (Perona et al., in preparation).. Conclusions The possibility of reconstructing a meaningful mechanistic model to reproduce the complex dynamics of alpine basins has been described in this work. Reconstructing the lumped input-outputs dynamics by exploring the traectories of the system in the phase space allows the differential model to experience the effects induced by noise in the inputs directly. This makes it robust and attractive with regard to future speculative and applied works. Speculative will aim at characterizing the system under a more dynamical viewpoint; applied works will look at the possibility of assessing the future system behaviour, as well as transferring the present approach to other alpine basins. References Corripio, J. G., and R.S. Purves, : Surface energy balance of high altitude glaciers in the Central Andes: the effect of snow penitents, in Climate and Hydrology in Mountain Areas, Edited by de Jong, C., Collins, D, Ranzi, R., John Wiley & Sons, -27. Dingman, S. L. Physical hydrology, Second Edition, Prentice Hall, New Jersey, 2; Eisenhammer, T., A. Hübler, N. Packard, and J.A.S. Kelso, 1991: Modeling experimental time series with ordinary differential equations, Biological cybernetics, 6, Ohmura, A., 1: Physical basis for the temperature-based melt-index method, Journal of Applied Meteorology, 4, Perona, P., A. Porporato, and L. Ridolfi, : On the Traectory Method for the reconstruction of differential equations from time series. Nonlinear Dynamics, 23, Saltzman, B., 1983: Climatic system analysis, Advances in Geophysics, 2,